# Properties

 Label 1024.2.e.d Level $1024$ Weight $2$ Character orbit 1024.e Analytic conductor $8.177$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1024 = 2^{10}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1024.e (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.17668116698$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 512) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{3} + ( - 2 i - 2) q^{5} + 4 i q^{7} + i q^{9}+O(q^{10})$$ q + (-i + 1) * q^3 + (-2*i - 2) * q^5 + 4*i * q^7 + i * q^9 $$q + ( - i + 1) q^{3} + ( - 2 i - 2) q^{5} + 4 i q^{7} + i q^{9} + (i + 1) q^{11} + ( - 2 i + 2) q^{13} - 4 q^{15} + 4 q^{17} + ( - 5 i + 5) q^{19} + (4 i + 4) q^{21} - 4 i q^{23} + 3 i q^{25} + (4 i + 4) q^{27} + (6 i - 6) q^{29} + 8 q^{31} + 2 q^{33} + ( - 8 i + 8) q^{35} + (2 i + 2) q^{37} - 4 i q^{39} - 2 i q^{41} + (3 i + 3) q^{43} + ( - 2 i + 2) q^{45} - 9 q^{49} + ( - 4 i + 4) q^{51} + ( - 2 i - 2) q^{53} - 4 i q^{55} - 10 i q^{57} + ( - 3 i - 3) q^{59} + ( - 6 i + 6) q^{61} - 4 q^{63} - 8 q^{65} + ( - 3 i + 3) q^{67} + ( - 4 i - 4) q^{69} - 4 i q^{71} + 4 i q^{73} + (3 i + 3) q^{75} + (4 i - 4) q^{77} + 8 q^{79} + 5 q^{81} + ( - 7 i + 7) q^{83} + ( - 8 i - 8) q^{85} + 12 i q^{87} + 12 i q^{89} + (8 i + 8) q^{91} + ( - 8 i + 8) q^{93} - 20 q^{95} - 4 q^{97} + (i - 1) q^{99} +O(q^{100})$$ q + (-i + 1) * q^3 + (-2*i - 2) * q^5 + 4*i * q^7 + i * q^9 + (i + 1) * q^11 + (-2*i + 2) * q^13 - 4 * q^15 + 4 * q^17 + (-5*i + 5) * q^19 + (4*i + 4) * q^21 - 4*i * q^23 + 3*i * q^25 + (4*i + 4) * q^27 + (6*i - 6) * q^29 + 8 * q^31 + 2 * q^33 + (-8*i + 8) * q^35 + (2*i + 2) * q^37 - 4*i * q^39 - 2*i * q^41 + (3*i + 3) * q^43 + (-2*i + 2) * q^45 - 9 * q^49 + (-4*i + 4) * q^51 + (-2*i - 2) * q^53 - 4*i * q^55 - 10*i * q^57 + (-3*i - 3) * q^59 + (-6*i + 6) * q^61 - 4 * q^63 - 8 * q^65 + (-3*i + 3) * q^67 + (-4*i - 4) * q^69 - 4*i * q^71 + 4*i * q^73 + (3*i + 3) * q^75 + (4*i - 4) * q^77 + 8 * q^79 + 5 * q^81 + (-7*i + 7) * q^83 + (-8*i - 8) * q^85 + 12*i * q^87 + 12*i * q^89 + (8*i + 8) * q^91 + (-8*i + 8) * q^93 - 20 * q^95 - 4 * q^97 + (i - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 4 q^{5}+O(q^{10})$$ 2 * q + 2 * q^3 - 4 * q^5 $$2 q + 2 q^{3} - 4 q^{5} + 2 q^{11} + 4 q^{13} - 8 q^{15} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 8 q^{27} - 12 q^{29} + 16 q^{31} + 4 q^{33} + 16 q^{35} + 4 q^{37} + 6 q^{43} + 4 q^{45} - 18 q^{49} + 8 q^{51} - 4 q^{53} - 6 q^{59} + 12 q^{61} - 8 q^{63} - 16 q^{65} + 6 q^{67} - 8 q^{69} + 6 q^{75} - 8 q^{77} + 16 q^{79} + 10 q^{81} + 14 q^{83} - 16 q^{85} + 16 q^{91} + 16 q^{93} - 40 q^{95} - 8 q^{97} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^11 + 4 * q^13 - 8 * q^15 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 8 * q^27 - 12 * q^29 + 16 * q^31 + 4 * q^33 + 16 * q^35 + 4 * q^37 + 6 * q^43 + 4 * q^45 - 18 * q^49 + 8 * q^51 - 4 * q^53 - 6 * q^59 + 12 * q^61 - 8 * q^63 - 16 * q^65 + 6 * q^67 - 8 * q^69 + 6 * q^75 - 8 * q^77 + 16 * q^79 + 10 * q^81 + 14 * q^83 - 16 * q^85 + 16 * q^91 + 16 * q^93 - 40 * q^95 - 8 * q^97 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$1023$$ $$\chi(n)$$ $$i$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 − 1.00000i 1.00000i
0 1.00000 + 1.00000i 0 −2.00000 + 2.00000i 0 4.00000i 0 1.00000i 0
769.1 0 1.00000 1.00000i 0 −2.00000 2.00000i 0 4.00000i 0 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.2.e.d 2
4.b odd 2 1 1024.2.e.b 2
8.b even 2 1 1024.2.e.c 2
8.d odd 2 1 1024.2.e.e 2
16.e even 4 1 1024.2.e.c 2
16.e even 4 1 inner 1024.2.e.d 2
16.f odd 4 1 1024.2.e.b 2
16.f odd 4 1 1024.2.e.e 2
32.g even 8 2 512.2.a.d yes 2
32.g even 8 2 512.2.b.a 2
32.h odd 8 2 512.2.a.c 2
32.h odd 8 2 512.2.b.b 2
96.o even 8 2 4608.2.a.f 2
96.o even 8 2 4608.2.d.b 2
96.p odd 8 2 4608.2.a.m 2
96.p odd 8 2 4608.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.c 2 32.h odd 8 2
512.2.a.d yes 2 32.g even 8 2
512.2.b.a 2 32.g even 8 2
512.2.b.b 2 32.h odd 8 2
1024.2.e.b 2 4.b odd 2 1
1024.2.e.b 2 16.f odd 4 1
1024.2.e.c 2 8.b even 2 1
1024.2.e.c 2 16.e even 4 1
1024.2.e.d 2 1.a even 1 1 trivial
1024.2.e.d 2 16.e even 4 1 inner
1024.2.e.e 2 8.d odd 2 1
1024.2.e.e 2 16.f odd 4 1
4608.2.a.f 2 96.o even 8 2
4608.2.a.m 2 96.p odd 8 2
4608.2.d.a 2 96.p odd 8 2
4608.2.d.b 2 96.o even 8 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1024, [\chi])$$:

 $$T_{3}^{2} - 2T_{3} + 2$$ T3^2 - 2*T3 + 2 $$T_{5}^{2} + 4T_{5} + 8$$ T5^2 + 4*T5 + 8 $$T_{47}$$ T47

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T + 2$$
$5$ $$T^{2} + 4T + 8$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} - 2T + 2$$
$13$ $$T^{2} - 4T + 8$$
$17$ $$(T - 4)^{2}$$
$19$ $$T^{2} - 10T + 50$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2} + 12T + 72$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} - 4T + 8$$
$41$ $$T^{2} + 4$$
$43$ $$T^{2} - 6T + 18$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 4T + 8$$
$59$ $$T^{2} + 6T + 18$$
$61$ $$T^{2} - 12T + 72$$
$67$ $$T^{2} - 6T + 18$$
$71$ $$T^{2} + 16$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 14T + 98$$
$89$ $$T^{2} + 144$$
$97$ $$(T + 4)^{2}$$